# Feedback parameters for a closed-loop multiple-input multiple-output model of the upper limb

## TL;DR

This paper introduces a method to estimate feedback parameters for a large-scale model of the upper limb's neuromusculoskeletal system, enabling better simulation of motor control.

## Contribution

The paper presents a novel method for estimating feedback gains and delays for a closed-loop MIMO model of the upper limb.

## Key findings

- Estimated feedback gains and delays were derived for a linear model of the upper limb with 13 muscles and 7 joint degrees of freedom.
- Validation showed correct sign of muscle-spindle feedback gains in all 39 tested muscle pairs.
- Efferent delay estimates showed a strong fit (R = 0.88) with measured innervation lengths.

## Abstract

Both closed-loop models and multi-input multi-output (MIMO) models of the neuromusculoskeletal system of the upper limb are important for simulating and understanding motor control. Yet no large-scale linear neuromusculoskeletal models of the upper limb that are both closed-loop and MIMO have been developed. The primary difficulty in creating such models is choosing appropriate feedback parameters (such as feedback gains and delays), as such a collection of parameters is not available in the literature. The purpose of this work is to 1) present a method for developing MIMO models of short-loop afferent feedback and 2) offer estimates of average feedback parameter values and ranges based on the currently available literature. To this end, we combined measurements of feedback-related parameters available in 26 prior studies with known properties of system stability and behavior. As a result, we present estimated feedback gains and delays for a linear model of the upper limb with inputs into the 13 major superficial muscles and outputs to the 7 main joint degrees of freedom from the shoulder to the wrist. This model includes homonymous feedback mediated by Golgi tendon organs and both homonymous and heteronymous feedback mediated by muscle spindles. As a partial validation of muscle-spindle feedback gains, we compared the sign of the estimated gains to known differences in excess central delay between excitatory and inhibitory connections. The comparison proved correct in all 39 muscle pairs for which we had both estimated a feedback gain and found a measured excess central delay value in the literature. Furthermore, as a partial validation of delay times, we compared estimated delay times to measured innervation lengths. We found a strong fit for efferent delays (R = 0.88) and a moderate fit for afferent delays (R = 0.65). In addition, we demonstrate the effect of feedback on model behavior and present brief comparisons between this behavior and experimentally observed behaviors of the human upper limb with and without feedback.

Computational modeling is an essential tool for analyzing upper-limb behavior; yet representing neural feedback in a large-scale model of the upper limb has proven very challenging. When considering neural feedback connections between n muscles, there are n2 possible connections to consider for each feedback mechanism, and modeling each feedback connection may require multiple feedback parameters (such as gain and delay values). A collection of feedback gains and propagation delays for each of these connections is not yet available in the literature. The purpose of this work is to 1) present a method for developing large-scale linear models of the upper-limb that include neural feedback and 2) offer estimates of average feedback parameter values and ranges based on the currently available literature. To accomplish these objectives, feedback-related parameters measured in 26 prior studies were combined with known properties of neuromusculoskeletal system behavior, resulting in a collection of parameter estimates for a linear model of the upper limb that includes the 13 major superficial muscles and the 7 main joint degrees of freedom from the shoulder to the wrist. Lastly, the simulated effects of neural feedback on system behavior were demonstrated.